1131 T Question 1

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1 1131 T Question 1 ( aks) You ae cycling, on a long staight path, at a constant speed of 6.0.s 1. Anothe cyclist passes you, taelling on the sae path in the sae diection as you, at a constant speed of b = 9.0.s 1. At the instant when she passes you, you ealise that she is a fiend of yous and you acceleate to catch up to he. You acceleate, stating fo t = 0, the tie when she passes you, with constant acceleation a = 1.5.s 2. a) i) On a displaceent-tie gaph, sketch you position, x a (t), and the position of you fiend, x b (t) as functions of tie, fo tie t < 0 (i.e. while she is still behind you). Label these sections of the gaphs x a and x b. ii) Also sketch displaceent-tie gaphs fo you you fiend fo tie t > 0. Label these sections of the gaphs x a and x b as well. iii) Showing you woking, deie both algebaic expessions and quantitatie alues fo the tie and distance it takes you to catch up with you fiend. b) In pat (a), you acceleated with constant acceleation and oetook he. In this pat, you acceleate with fowads acceleation a = 1.5.s 2 fo a tie T 1, then deceleate with fowads acceleation a = 1.5.s 2 fo a tie T 2. You judge T 1 and T 2 so that when you stop deceleating, you ae taelling alongside he at the sae speed. Daw a second displaceent-tie gaph to show this situation. Clealy ak x a, x b and the tie inteals T 1 and T 2. c) You ae cycling Noth with elocity 1. Relatie to you, the wind appeas to be coing fo the East. You double you speed to 2 1, still in the Noth diection. The wind has not changed, but now it appeas to be coing diectly fo the Noth East (ie at 45 ahead of you and fo the ight). i) Deie an expesion fo the speed of the wind, with espect to the gound, in tes of 1. ii) What is the diection of the wind with espect to the gound?

2 Question 1 iii) x b = b t x x a x a = 0 t at2 catch up when x a = x b b t = 0 t at2 x b b 0 = 1 2 at t = 2( b 0 ) a x a b) x b t = 2(9.0.s s 1 ) 1.5.s 2 = 4.0 s. Distance taelled in that tie: x b = b t = 2 b( b 0 ) a = 36. x x b x a T 2 x a T 1 t x b

3 c) Let the appaent elocity of the wind be ' w in the fist case and " w in the second case. Its elocity oe the gound is w. w = 1 + ' w and siilaly fo second case w = " w Coodinates fist case second case N o y E o x 1 ' w w 1 1 w " w 45 Hee we note that the top tiangle in the RH diaga is the sae as tiangle fo case 1. We also see that the lage tiangle at ight is an isosceles tiangle. So the answe to (ii) is that the wind coes fo the SW. i) The speed of the wind, fo any of these ight angle tiangles: 1 / w = sin 45 so w = 2 1. ii) The wind coes fo the SE, ie fo 45 South of East (o it is going NW, o any equialent stateent. (No ebal justification sought.)

4 Question 2 ( aks) i) Wite Newton's second law in a fo that applies to a finite object that is not necessaily igid, but that has constant ass. If you stateent is an equation, state caefully the eaning of each te. (Fo exaple, do not let the ake wonde "what foce?" o "what acceleation?".) The sketches show successie states of a an juping etically in the ai. He begins (sketch A) fo a stationay position with his legs bent. He then staightens his legs and ankles apidly: sketch B shows the oent at which his feet leae the gound. The lines aked "CoM" show the height of his cente of ass. Between the fist two sketches, his cente of ass ises a distance L. Sketch C shows hi at the point whee his cente of ass has its axiu height, which is a etical distance h aboe its height at the point of take-off. You ay neglect ai esistance. A B CoM h CoM L CoM ii) Showing you woking, and stating any assuptions you ake, deteine the speed of the an's cente of ass at the oent (B) when his feet leae the gound. iii) Assue that the etical acceleation a c of his cente of ass is constant between A and B. Deie an expession fo a c. i) Using you answe to pat (i), and thinking caefully, deie an expession fo the etical foce N exeted by the gound on his feet duing the phase A to B. ) If the an's ass is 70 kg, if L is 0.4 and h is 0.6, what is the downwads foce (assued constant) exeted by his feet duing the phase A to B? State any physical law o pinciple you use in obtaining you answe.

5 Question 2 i) Σ F ext =.a c whee Σ F ext is the total extenal foce, is the ass of the object and a c is the acceleation of the cente of ass of the object OR F ext =.a c whee F ext is the total extenal foce, is the ass of the object and a c is the acceleation of the cente of ass of the object ii) OR Σ F ext = d dt (. c) OR Σ F ext = d dt p c etc duing the jup phase (B C), no extenal nonconseatie foces act, so the echanical enegy associated with the cente of ass is conseed U i + K i = U f + K f o U + K = 0 At C, K = 0, so 1 2 c 2 = U ga = gh iii) i) so c = 2gh so c 2 = 2gh otion in one diension with constant acceleation: 2a y L = yf 2 yi 2. othe notations acceptable 2a c L = c 2 a c = c 2 2L = gh L Σ F ext =.a c applied in the etical diection gies N g =.a c N = (g + a c ) = g 1 + h L OR N = g 1 + h L up. (diection is upwads) ) Fo Newton's thid law, this foce F has agnitude N but is downwads. F = g 1 + h L (down). F = 1.7 kn.

6 Question 3 a) ω A sall, flat agnet, of ass, is positioned at a distance fo the cente of a steel disc that otates with angula elocity ω about a hoizontal axis, as shown. The agnitude of the agnetic foce between the agnet and the disc is F and it is in the noal diection only. The coefficients of static and kinetic fiction between the disc and the agnet ae espectiely µ s and µ k espectiely, and µ s > µ k. The agnet does not slide when the disc is stationay. What is the axiu alue of ω at which the agnet will not slide on the disk? (Hint: at which point is it ost likely to begin to slide?) b) With espect to a ey lage sepaation, the potential enegy of a pai of asses M and sepaated by is U = GM/. The agnitude of the gaitational foce between the is F = GM/ 2, whee is the distance between thei centes and whee G is the uniesal constant of gaitation. Deteine the total echanical enegy E of a sall satellite (ass ) in a cicula obit of adius aound a planet of ass M in tes of G, M, and. (Hint: what is the centipetal foce?) c) Using pats of you answe to (b) o othewise, deie a elation between the obital peiod T and the adius fo a cicula obit of a sall ass about a lage ass M. (Hint: what is the cicufeence?) d) A satellite, ass = 120 kg, will be be assebled in the intenational space station (ISS), which is in a cicula obit about the Eath, at an altitude (i.e. distance aboe the suface of the Eath) of 350 k. It is then to be oed to a geosynchonous obit, i.e. one in which it is always diectly aboe a paticula point on the equato. How uch enegy is equied to oe it fo the ISS to the geosynchonous obit? The adius of the Eath is 6,400 k, its ass is 6.0 x kg and G = 6.67 x N. 2 kg -2.

7 Question 3 While it is not sliding, the agnet undegoes unifo cicula otion, so the total foce on it is a c = ω 2 towads the cente. The total foce is F f + g. The geatest fictional foce is equied at the botto of the cicle, whee, taking the upwads diection as positie: F f g = ΣF = a c = ω 2 so F f = g + ω 2 (At the top of the cicle, F f g = ω 2 so F f = ω 2 g and at inteediate angles it has inteediate alues.) Fo the definition of static fiction, the axiu alue of F fax = µ s N, whee N is the noal foce, so F fax = µ s N = g + ω ax 2 Hee, the only foce in the noal diection is F, so F = N, µ s F = g + ω ax 2 ω ax 2 = µ s F g ω ax = µ s F g b) E = U + K K = centipital foce = 2 = F g = so 2 = GM so E = U + K = GM + 1 GM 2 = GM 2 c) Aboe we had 2 = GM The obit cicufeence is 2π, so = 2π/T, so 2π T 2 GM = 4π 2 3 = GMT 2 GM 2 (o equialent) d) Fo a geosynchonous obit, the peiod T (= 23.9 hous) 24 hous. Fo (c) = GMT2 1/3 4π 2 Substitution gies geosynch = 42,000 k. Fo (b) E = GM f =... = 3.0 GJ i [This answe will ean full aks. Howee, it should be noted that, in pactice, a uch geate quantity of enegy will be expended, ost of used to oe fuel and the containe fo that fuel.]

8 Question 4 L θ o L Two light, inextensible stings, each of length L ae hung fo the sae fixed point, as shown. On one, a lup of plasticine (a soft ateial) of ass M is attached. Initially, it hangs etically. On the othe sting, a ball of ass is attached. Initially, it is stationay, but displaced so that its sting akes and angle θ o with the etical, as shown. The diensions of the ball and plasticene ae uch salle than L. The ball is then eleased. When its sting is etical, it stikes the plasticine and the two eain stuck togethe. Showing all woking and stating any assuptions you ake, deie an expession fo the speed of the cobined object, iediately afte the collision, in tes of the paaetes gien and g, the gaitational acceleation. Ai esistance is negligible. M Question 4 a b c d θ o L L α Befoe the collision (a to b), no nonconseatie foces act, so echanical enegy is conseed. Taking the botto of the path as the zeo fo U: U a + K a = U b + K b gh + 0 = = 2gh = 2gL(1 cos θ) = 2gL(1 cos θ). h M M M+ V H Duing the collision (b to c), extenal foces in the hoizontal diection ae negligible, so oentu is conseed in the hoizontal diection. So + 0 = (M+)V V = M+ = M+ 2gL(1 cos θ). M+

9 Question 5 Two cylindical jas each hae adius. The thickness of the walls is negligible copaed with. When epty, the ass of each ja is and thei adius of gyation k = (to an appoxiation sufficient fo this poble). One ja is full of wate with ass M. The othe is full of honey with ass M. They ae both placed, stationay, on an inclined plane aking an angle θ with the hoizontal. Thei oientation on the plane allows the to oll to the botto along the shotest path on the plane. Fiction between jas and plane is always sufficient to ensue olling. i) The iscosity of the wate is sufficiently low that the wate does not otate. Deteine the speed of the ja of wate afte it has olled a distance s down the plane. ii) The iscosity of the honey is sufficiently high that the honey otates as a solid object, at the sae ate as the ja. Deteine the speed of the ja of honey afte it has olled a distance s down the plane. The oent of inetia of a hoop is 2. That of a disc is 1 2 2, whee tes hae thei usual eaning. Hint: what is the elatie elocity at the point of contact duing olling? Question 5 = 0, ω = 0 h s θ ω θ Because the elatie elocity at the point of contact duing olling is zeo, nonconseatie foces do no wok, theefoe echanical enegy is conseed. U + K = 0 K = U 1 2 (+M) Iω2 = (M+)gh Rolling ω = / so (+M) 2 + I 2 / 2 = 2(M+)gh i) Fo wate, only the ja otates, so I = k 2 = 2, so (+M) / 2 = 2(M+)gh (2+M) 2 = 2(M+)gh = gh 2(M+) 2+M = gh 1+/M 1/2+/M ii) Fo honey, both ja and contents otate, so I = M2 so (+M) 2 + ( M)2 2 / 2 = 2(M+)gh (2+3M/2) 2 = 2(M+)gh = gh 2(M+) 2+3M/2 = gh 1+/M 3/4+/M

1121 T Question 1

1121 T Question 1 1121 T1 2008 Question 1 ( aks) You ae cycling, on a long staight path, at a constant speed of 6.0.s 1. Anothe cyclist passes you, tavelling on the sae path in the sae diection as you, at a constant speed